The distributive property states that multiplying the sum of two numbers by another number, is equivalent to multiplying each of the addends by that number and then adding the products together.

The Distributive Property stands for the product of a number where the sum is equal to the sum of the individual products of the addends and the number.

1. a * (b + c) = (a * b) + (a * c)

2. 3 * (4 + 5) = (3 * 4) + (3 * 5)

## Distributive Property Definition

The distributive property states that a common factor can be factored out or it may also be stated as "a factor can be distributed over a sum either from the right or from the left".

For any three real numbers a, b and c the distributive property says a(b+c) = ab + ac. By using the property of Symmetry, we can rewrite it as ab + ac = a(b+c)

This latter form of the Distributive Property, ab + ac = a(b+c), is often used to simplify polynomials.

Example: 4x2 + 7x2 is a polynomial which can be simplified by using the distributive property as

4x2 + 7x2 = (4+7)x2

= 11x2

The algebraic terms with same variable raised to the same power are like terms.
We have combined the like terms. The unlike terms cannot be combined. The result which is simplified represents the same real number as the original expression when the variable or variables are replaced by real numbers.

The expressions that represent the same real numbers when variables are replaced by real numbers are called as Equivalent Expressions.

There a few forms of the Distributive Property that are necessary to simplify the polynomial expressions

1. -(a + b) = -1(a + b) = -a - b
2. -(a - b) = -1(a - b) = -a + b
3. (a + b)(c + d) = a(c + d) + b(c + d)
The like terms within any bracket must be combined as far as possible before the brackets are removed.

Again ,If a, b and c are any three integers, then,

a (b + c) = (a × b) + (a × c) [right distributive law]

(b + c) a = (b x a) + (c x a) [left distributive law]

In the left hand side of the above equation, ‘a' multiplies with the sum of ‘b' and ‘c' on the right hand side, it multiplies ‘b' and the ‘c' individually, with the results added afterwards. Because both expression give the same value, we can say that multiplication by ‘a' distributes over the addition of b and c. Since if we put any integer in place of ‘a', ‘b', and ‘c' above, and still have the same value of the expression, we say that the multiplication of integers distributes over the addition of integers.

Consider the following:

1) 5 (2 + 3) = (5 × 2) + (5 × 3)

Consider the left hand side (L.H.S), 5 (2 + 3) [Here the left hand side of the equation, 5 multiplies the sum of 2 and 3]

= 5(5)

= 25

Now consider right hand side (R.H.S), (5 × 2) + (5 × 3) [here we observe that in the right hand side, 5 multiplies 2 and 3 individually, with the results added afterwards].

= (10) + (15) = 25

Therefore, L.H.S = R.H.S

2) 4 (1 + 3) = (4 × 1) + (4 × 3)

Consider the left hand side (L.H.S), 4 (1 + 3) [Here in the left hand side of the equation, 4 multiplies the sum of 1 and 3]

= 4(4)

= 16

Now consider right hand side (R.H.S), (4 × 1) + (4 × 3) [here we observe that in the right hand side, 4 multiplies 1 and 3 individually, with the results added afterwards].

= (4) + (12) = 16

Therefore, L.H.S = R.H.S

## Factoring using the Distributive Property

The steps involved in factoring using the distributive property are
Step 1: Find the greatest common factor.
Step 2: Factor out the greatest common factor.
Step 3: Factor by grouping.
Step 4: Solve equations by factoring.
Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping. It is not always obvious which terms to group, and sometimes several different groupings will work.

### Examples on Factoring using the Distributive Property

Example 1: Use factoring by grouping to factor x3 - 2x2 - 3x + 6

Solution : x3 - 2x2 - 3x + 6 = (x3 - 2x2) - (3x - 6) [Group terms]
= x2 (x - 2) - 3(x - 2) [Factor each group]
= (x - 2)(x2 - 3) [Distributive property]

Example 2: Use factoring by grouping to factor 7p2 - 4pq + 12 q4 - 21pq3

Solution : 7p2 - 4pq + 12 q4 - 21pq3 = (7p2 - 4pq) + (12q4 - 21pq3) [Group terms]
= p (7p - 4q) + 3q3(4q - 7p) [Factor each group]
= p (7p - 4q) - 3q3(7p - 4q)
= (p - 3q3)(7p - 4q) [Distributive property]

## Distributive Property of Multiplication over Addition

The distributive property of multiplication over addition states that when the sum of two numbers is multiplied by a number, then the first number is distributed to those two numbers and it is multiplied by each of them separately.
a x (b + c) = a x b + a x c

### Examples on distributed property of multiplication over addition

Example1: Solve 6 x (3 + 5) = 6 x 3 + 6 x 5
= 18 + 30
= 48
Example2: Solve 4 x (2 + 5) = 4 x 2 + 4 x 5
= 8 + 20
= 28
Example 3: Solve y x (3y + 7y) = y x 3y + y x 7y
= 3y2 + 7y2
= 10 y2
Example 4: Solve z2(4z + 5z) = z2 x 4z + z2 x 5z
= 4z3 + 5z3
= 9 z3

### Word Problem on distributed property of multiplication over addition

Example 1: John bought 6 sheets and a blanket. The sheets cost 17 dollars each and the blanket cost 8 dollars. What is John's total bill?
Solution: (17 $\times$ 6) + (8 $\times$ 6)
=102 + 48
= 150 dollars

Example 2: Rita bought 4 pens and pencils. The pens cost 3 dollars each and the pencils cost 2 dollars. Find Rita's total bill?
Solution: (3 $\times$ 4) + (2 $\times$ 4)
=12 + 8
= 20 dollars

## Distributive Property Problems

The following problems on distributive property will explain operations on distributive property like multiplication.

1. Apply the distributive property to the expression 2(x + 3) and then simplify the result.

Solution
= 2(x + 3) = 2(x) + 2(3) [Distributive property multiplication]
= 2x + 6

2. Apply the distributive property to the expression 5(2x - 8) and then simplify the result

Solution
= 5 (2x - 8) - 5 (2x) - 5 (8) [Distributive property multiplication]
= 10x – 40

3. Apply the distributive property to the expression 4 (2a + 3) + 8 and then simplify the result.

Solution
= 4 (2a + 3) + 8 = 4 (2a) + 4(3) + 8 [Distributive property multiplication and addition]
= 8a + 12 + 8
= 8a + 20

4. Apply the distributive property to the expression $\frac{1}{2}$ (3x +6) and then simplify the result.

Solution
= $\frac{1}{2}$ (3x +6) = $\frac{1}{2}$ (3x) + $\frac{1}{2}$ (6) [Distributive property of multiplication]
= ($\frac{3}{2}$)x + 3

5. Apply the distributive property to the expression 12 [ ($\frac{2}{3}$)x + ($\frac{1}{2}$)y ] and then simplify the result.

Solution
= 12 [ ($\frac{2}{3}$)x + ($\frac{1}{2}$)y ]
= 12($\frac{2}{3}$)x + 12($\frac{1}{2}$)y [Distributive property of multiplication]
= 8x + 6y

### Simplifying Expressions

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